Phase Coherence in Rössler's "Band"

In the case of the Lorenz attractor, a ball of nearby initial points quickly spreads over the entire attractor. But not all chaotic attractors are characterized by such rapid mixing. A good example is afforded by what is sometimes called "Rössler's band."⁽¹⁾ Here, initial conditions mix radially at a fair clip, but only very slowly azimuthally. In other words, nearby points on the attractor travel together around the attractor, even while spreading out radially. This is illustrated in the animation at the right. (Click HERE to start.) In this case, the geometry is quite simple. At the center of the attractor is a saddle focus of index 2. Trajectories in the vicinity of the focus spiral away from it, eventually rising out of the plane of the central disk, only to be re-injected down onto it by virtue of coming under the influence of the one-dimensional stable manifold. The phenomenological consequence of this geometry is phase coherence as manifested by "instrumentally" sharp peaks in the power spectrum.